⚛️ general relativity

Does spatial curvature generate new thermodynamic criticality at the FLRW apparent horizon?

This paper demonstrates that while non-zero spatial curvature in a FLRW universe shifts the values of thermodynamic critical quantities like temperature and pressure, it does not generate a new universality class, as the critical ratio and mean-field exponents remain unchanged under constant-curvature thermodynamic variations.

Original authors: Samuel Lepe, Joel Saavedra

Published 2026-09-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Samuel Lepe, Joel Saavedra

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The universe is often imagined as a vast, empty stage where galaxies drift apart, but to physicists, the cosmos is also a thermodynamic system, much like a pot of gas or a block of ice, governed by the same laws of heat and pressure that rule our everyday world. For decades, scientists have explored a surprising connection between gravity and thermodynamics, discovering that the boundaries of space, known as horizons, behave as if they possess a temperature and an entropy, a measure of disorder. In the standard model of cosmology, which describes a universe that is perfectly flat and uniform, researchers have already found that the edge of our observable universe can exhibit a strange, critical behavior similar to the point where water turns into steam. At this critical point, the distinction between different states of matter blurs, and the system becomes highly sensitive to tiny changes. However, our universe might not be perfectly flat; it could be slightly curved, like the surface of a sphere or a saddle. This raises a fundamental question: does this curvature change the rules of the game, creating entirely new types of cosmic phase transitions, or does it simply shift the numbers while leaving the underlying physics the same?

A team of researchers from the Pontificia Universidad Católica de Valparaíso in Chile has tackled this question by extending the thermodynamic description of the universe to include spatial curvature. They focused on a specific model where the universe contains cold, invisible matter and a mysterious form of energy that drives the expansion of space, known as holographic dark energy. In a flat universe, the relationship between the temperature and pressure at the cosmic horizon is straightforward, but when the universe is curved, an extra geometric variable enters the equation, making the standard mathematical tools ambiguous. To solve this, the researchers introduced a new way of looking at the problem. Instead of trying to track the universe as it evolves over time, they froze a specific aspect of the curvature and examined the thermodynamic behavior on these fixed slices. This allowed them to define a clear equation of state, a rule that links the temperature, pressure, and volume of the cosmic horizon, even when the universe is not flat.

Their investigation revealed that spatial curvature does not generate a new kind of thermodynamic interaction or a new class of critical behavior. Instead, it acts as a dial that rescales the existing critical point. When the universe is curved, the specific values for the critical temperature, pressure, and volume shift. For a closed universe, which curves back on itself like a sphere, the critical volume becomes smaller while the temperature and pressure needed to reach that critical state increase. Conversely, in an open universe that curves like a saddle, the critical volume grows larger, and the required temperature and pressure drop. Despite these shifts in scale, the fundamental nature of the transition remains unchanged. The ratio between the critical quantities and the mathematical exponents that describe how the system behaves near the critical point are identical to those found in a flat universe. In essence, the curvature changes the size of the stage and the intensity of the actors, but the play itself remains the same.

The researchers went further to map out the global structure of this phase transition, constructing a thermodynamic potential that describes the stability of the cosmic horizon. They identified regions where the universe could exist in two different states simultaneously, much like water and steam coexisting in a boiling pot. By calculating the energy of these competing states, they derived a precise curve that marks where one phase turns into the other. This curve, known as the coexistence line, connects smoothly to the critical point, where the distinction between the two phases disappears. They also identified the boundaries of stability, showing that if the universe were pushed too far from this equilibrium, it would become mechanically unstable. A key finding was that the energy required to switch between these phases, known as latent heat, vanishes exactly at the critical point, confirming that the transition is a continuous, global event within this theoretical framework.

However, the authors are careful to distinguish between this theoretical equilibrium and the actual history of our universe. The analysis was performed on fixed slices of curvature, but in a real, evolving universe, the curvature variable changes as the cosmos expands. Therefore, while the thermodynamic landscape contains a critical point and a path for phase transition, it is not guaranteed that our universe's trajectory will actually cross this path. Whether the universe ever reaches this critical state depends on the specific mix of matter and energy it contains and how it has evolved over billions of years. The study establishes that the potential for such a transition exists and defines its properties, but determining if it ever happens in reality requires solving the full dynamical equations of the cosmos. The work clarifies that spatial curvature, while influential, does not rewrite the fundamental laws of cosmic thermodynamics; it merely adjusts the scale at which they operate.

Technical Summary: Spatial Curvature and Thermodynamic Criticality at the FLRW Apparent Horizon

Problem Statement
Previous work established that a spatially flat FLRW universe containing non-interacting cold dark matter and holographic-type dark energy exhibits van der Waals-like thermodynamic criticality at the apparent horizon [1, 16]. However, extending this framework to universes with non-zero spatial curvature ( $k \neq 0$ ) presents a fundamental thermodynamic ambiguity. In the flat case ( $k=0$ ), the apparent horizon radius $R_A$ is uniquely determined by the Hubble parameter $H$ ( $R_A = 1/H$ ), allowing the dark energy density to be expressed solely as a function of the horizon radius. In contrast, for $k \neq 0$ , the relation $H^2 = 1/R_A^2 - k/a^2$ introduces the scale factor $a(t)$ as an independent geometric variable. Consequently, the thermodynamic pressure on the horizon becomes a function of temperature ( $T$ ), specific volume ( $v$ ), and the scale factor ( $a$ ), i.e., $P_{th} = P_{th}(T, v, a; k)$ . Without a closure prescription for the curvature sector, derivatives with respect to the specific volume are ill-defined, preventing the standard identification of critical points.

Methodology
The authors generalize the apparent-horizon thermodynamic construction to non-flat FLRW spacetimes by introducing a specific ensemble closure:

Dimensionless Curvature Variable: A dimensionless variable $\chi \equiv k R_A^2 / a^2$ is defined. This variable relates to the curvature density parameter $\Omega_k$ via $\chi = -\Omega_k / (1-\Omega_k)$ .
Fixed-Curvature Ensemble: The thermodynamic state space is restricted to slices of constant $\chi$ . This allows for well-defined partial derivatives $(\partial P_{th}/\partial v)_{T, \chi}$ and $(\partial^2 P_{th}/\partial v^2)_{T, \chi}$ , establishing a consistent two-dimensional thermodynamic ensemble.
Geometric Cancellation: The authors demonstrate that explicit curvature terms cancel out in the work density equation of state derived from the unified first law. The non-ideal interaction required for criticality arises solely from the holographic dark energy sector, not from the geometry itself.
Holographic Model: The dark energy density is modeled as $\rho_{de} = 3\alpha H^{2n}$ , where $n$ is a nonlinear power and $\alpha$ is a coupling constant.
Global Phase Analysis: To determine if local criticality implies a global first-order phase transition, the authors construct the Helmholtz potential $F$ and Gibbs free energy $G$ using the physical thermodynamic volume $V_A = \pi v^3/6$ and the Bekenstein-Hawking entropy. They apply the Maxwell equal-area construction to identify coexistence curves.

Key Contributions and Results

Renormalization of Critical Scales: For a fixed $\chi$ , the equation of state takes the form $P_{th} = 8T/v - 2/v^2 + C_\chi/v^{2n}$ , where $C_\chi$ is a coefficient dependent on $\chi$ . The authors find that spatial curvature shifts the dimensional critical quantities (specific volume $v_c$ , temperature $T_c$ , and pressure $P_c$ ) but leaves the reduced equation of state and the critical ratio $P_c v_c / T_c$ unchanged.
Universality Class Preservation: The critical exponents ( $\alpha_{crit}=0, \beta_{crit}=1/2, \gamma_{crit}=1, \delta_{crit}=3$ ) are derived and found to be independent of $\alpha$ , $\chi$ , and $n$ . These exponents satisfy standard mean-field scaling relations. Thus, spatial curvature does not generate a new local universality class; it merely rescales the critical point.
Global Coexistence and Latent Heat: For the quadratic holographic model ( $n=2$ ), the authors derive an exact parametric coexistence curve. They show that the equality of Gibbs free energies between competing branches is equivalent to the Maxwell construction in the $P_{th}$ - $V_A$ plane. The latent heat associated with the phase transition vanishes continuously at the critical endpoint, confirming a global first-order coexistence structure within the fixed-curvature ensemble.
Spinodal Structure: The limits of local mechanical stability (spinodal branches) are identified. The region between the spinodal curves is mechanically unstable, while the regions between the coexistence curve and the spinodal branches represent metastable phases.

Significance and Claims
The paper claims that spatial curvature acts as a renormalization factor for the critical scales of the apparent horizon thermodynamics but does not alter the fundamental nature of the phase transition or the universality class. The critical behavior is driven by the nonlinear holographic dark energy sector, not the curvature itself.

Crucially, the authors distinguish between the thermodynamic ensemble and cosmological evolution. The fixed- $\chi$ analysis defines a valid equilibrium state space, but a physical FLRW trajectory generally involves time-evolving $T$ , $v$ , and $\chi$ . Therefore, the existence of a critical point in the fixed-curvature ensemble does not guarantee that a physical universe will intersect this critical locus. Determining whether a specific cosmological solution crosses the critical or coexistence lines requires solving the full dynamical Friedmann equations, a separate question from the thermodynamic existence of the critical point. Furthermore, the authors note that even if a trajectory intersects the coexistence surface, a physical phase transition would require the relaxation time of the system to be shorter than the cosmological expansion timescale, a condition not addressed by the equilibrium equation of state alone.

Technical Summary: Spatial Curvature and Thermodynamic Criticality at the FLRW Apparent Horizon

More like this